Biostat 212a Homework 6

Due Mar 22, 2024 @ 11:59PM

Author

Yangn An and UID: 106332601

Published

March 21, 2024

Load R libraries.

rm(list = ls())
library(tidyverse)
library(tidymodels)
library(readr)
library(tswge)
library(ggplot2)
library(yardstick)

acfdf <- function(vec) {
    vacf <- acf(vec, plot = F)
    with(vacf, data.frame(lag, acf))
}

ggacf <- function(vec) {
    ac <- acfdf(vec)
    ggplot(data = ac, aes(x = lag, y = acf)) + geom_hline(aes(yintercept = 0)) + 
        geom_segment(mapping = aes(xend = lag, yend = 0))
}

tplot <- function(vec) {
    df <- data.frame(X = vec, t = seq_along(vec))
    ggplot(data = df, aes(x = t, y = X)) + geom_line()
}

1 New York Stock Exchange (NYSE) data (1962-1986) (140 pts)

Figure 1: Historical trading statistics from the New York Stock Exchange. Daily values of the normalized log trading volume, DJIA return, and log volatility are shown for a 24-year period from 1962-1986. We wish to predict trading volume on any day, given the history on all earlier days. To the left of the red bar (January 2, 1980) is training data, and to the right test data.

The NYSE.csv file contains three daily time series from the New York Stock Exchange (NYSE) for the period Dec 3, 1962-Dec 31, 1986 (6,051 trading days).

  • Log trading volume (\(v_t\)): This is the fraction of all outstanding shares that are traded on that day, relative to a 100-day moving average of past turnover, on the log scale.

  • Dow Jones return (\(r_t\)): This is the difference between the log of the Dow Jones Industrial Index on consecutive trading days.

  • Log volatility (\(z_t\)): This is based on the absolute values of daily price movements.

# Read in NYSE data from url

url = "https://raw.githubusercontent.com/ucla-biostat-212a/2024winter/master/slides/data/NYSE.csv"
NYSE <- read_csv(url)

NYSE
# A tibble: 6,051 × 6
   date       day_of_week DJ_return log_volume log_volatility train
   <date>     <chr>           <dbl>      <dbl>          <dbl> <lgl>
 1 1962-12-03 mon         -0.00446     0.0326           -13.1 TRUE 
 2 1962-12-04 tues         0.00781     0.346            -11.7 TRUE 
 3 1962-12-05 wed          0.00384     0.525            -11.7 TRUE 
 4 1962-12-06 thur        -0.00346     0.210            -11.6 TRUE 
 5 1962-12-07 fri          0.000568    0.0442           -11.7 TRUE 
 6 1962-12-10 mon         -0.0108      0.133            -10.9 TRUE 
 7 1962-12-11 tues         0.000124   -0.0115           -11.0 TRUE 
 8 1962-12-12 wed          0.00336     0.00161          -11.0 TRUE 
 9 1962-12-13 thur        -0.00330    -0.106            -11.0 TRUE 
10 1962-12-14 fri          0.00447    -0.138            -11.0 TRUE 
# ℹ 6,041 more rows

The autocorrelation at lag \(\ell\) is the correlation of all pairs \((v_t, v_{t-\ell})\) that are \(\ell\) trading days apart. These sizable correlations give us confidence that past values will be helpful in predicting the future.

Code 
ggacf(NYSE$log_volume) + ggthemes::theme_few()

Figure 2: The autocorrelation function for log volume. We see that nearby values are fairly strongly correlated, with correlations above 0.2 as far as 20 days apart.

Do a similar plot for (1) the correlation between \(v_t\) and lag \(\ell\) Dow Jones return \(r_{t-\ell}\) and (2) correlation between \(v_t\) and lag \(\ell\) Log volatility \(z_{t-\ell}\).

Code 
seq(1, 30) %>% 
  map(function(x) {cor(NYSE$log_volume , lag(NYSE$DJ_return, x), use = "pairwise.complete.obs")}) %>% 
  unlist() %>% 
  tibble(lag = 1:30, cor = .) %>% 
  ggplot(aes(x = lag, y = cor)) + 
  geom_hline(aes(yintercept = 0)) + 
  geom_segment(mapping = aes(xend = lag, yend = 0)) + 
  ggtitle("AutoCorrelation between `log volume` and lagged `DJ return`")

Figure 3: Correlations between log_volume and lagged DJ_return.
Code 
seq(1, 30) %>% 
  map(function(x) {cor(NYSE$log_volume , lag(NYSE$log_volatility, x), use = "pairwise.complete.obs")}) %>% 
  unlist() %>% 
  tibble(lag = 1:30, cor = .) %>% 
  ggplot(aes(x = lag, y = cor)) + 
  geom_hline(aes(yintercept = 0)) + 
  geom_segment(mapping = aes(xend = lag, yend = 0)) + 
  ggtitle("AutoCorrelation between `log volume` and lagged `log volatility`")

Figure 4: Weak correlations between log_volume and lagged log_volatility.

1.1 Project goal

Our goal is to forecast daily Log trading volume, using various machine learning algorithms we learnt in this class.

The data set is already split into train (before Jan 1st, 1980, \(n_{\text{train}} = 4,281\)) and test (after Jan 1st, 1980, \(n_{\text{test}} = 1,770\)) sets.

In general, we will tune the lag \(L\) to acheive best forecasting performance. In this project, we would fix \(L=5\). That is we always use the previous five trading days’ data to forecast today’s log trading volume.

Pay attention to the nuance of splitting time series data for cross validation. Study and use the time-series functionality in tidymodels. Make sure to use the same splits when tuning different machine learning algorithms.

Use the \(R^2\) between forecast and actual values as the cross validation and test evaluation criterion.

1.2 Baseline method (20 pts)

We use the straw man (use yesterday’s value of log trading volume to predict that of today) as the baseline method. Evaluate the \(R^2\) of this method on the test data.

Before we tune different machine learning methods, let’s first separate into the test and non-test sets. We drop the first 5 trading days which lack some lagged variables.

# Lag: look back L trading days
# Do not need to include, as we included them in receipe
L = 5

for(i in seq(1, L)) {
  NYSE <- NYSE %>% 
    mutate(!!paste("DJ_return_lag", i, sep = "") := lag(NYSE$DJ_return, i),
           !!paste("log_volume_lag", i, sep = "") := lag(NYSE$log_volume, i),
           !!paste("log_volatility_lag", i, sep = "") := lag(NYSE$log_volatility, i))
}


NYSE <-   NYSE %>% na.omit()
# Drop beginning trading days which lack some lagged variables
NYSE_other <- NYSE %>% 
  filter(train == 'TRUE') %>%
  select(-train) %>%
  drop_na()
dim(NYSE_other)
[1] 4276   20
NYSE_test = NYSE %>% 
  filter(train == 'FALSE') %>%
  select(-train) %>%
  drop_na()
dim(NYSE_test)
[1] 1770   20
library(yardstick)
# cor(NYSE_test$log_volume, NYSE_test$log_volume_lag1) %>% round(2)
r2_test_strawman =  rsq_vec(NYSE_test$log_volume, lag(NYSE_test$log_volume, 1)) %>% round(2)
print(paste("Straw man test R2: ", r2_test_strawman))
[1] "Straw man test R2:  0.35"

1.3 Autoregression (AR) forecaster (30 pts)

  • Let \[ y = \begin{pmatrix} v_{L+1} \\ v_{L+2} \\ v_{L+3} \\ \vdots \\ v_T \end{pmatrix}, \quad M = \begin{pmatrix} 1 & v_L & v_{L-1} & \cdots & v_1 \\ 1 & v_{L+1} & v_{L} & \cdots & v_2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & v_{T-1} & v_{T-2} & \cdots & v_{T-L} \end{pmatrix}. \]

  • Fit an ordinary least squares (OLS) regression of \(y\) on \(M\), giving \[ \hat v_t = \hat \beta_0 + \hat \beta_1 v_{t-1} + \hat \beta_2 v_{t-2} + \cdots + \hat \beta_L v_{t-L}, \] known as an order-\(L\) autoregression model or AR(\(L\)).

  • Before we start the model training, let’s talk about time series resampling. We will use the rolling_origin function in the rsample package to create a time series cross-validation plan.

  • When the data have a strong time component, a resampling method should support modeling to estimate seasonal and other temporal trends within the data. A technique that randomly samples values from the training set can disrupt the model’s ability to estimate these patterns.

NYSE %>% 
  ggplot(aes(x = date, y = log_volume)) + 
  geom_line() + 
  geom_smooth(method = "lm")

wrong_split <- initial_split(NYSE_other)

bind_rows(
  training(wrong_split) %>% mutate(type = "train"),
  testing(wrong_split) %>% mutate(type = "test")
) %>% 
  ggplot(aes(x = date, y = log_volume, color = type, group = NA)) + 
  geom_line()

correct_split <- initial_time_split(NYSE_other %>% arrange(date))

bind_rows(
  training(correct_split) %>% mutate(type = "train"),
  testing(correct_split) %>% mutate(type = "test")
) %>% 
  ggplot(aes(x = date, y = log_volume, color = type, group = NA)) + 
  geom_line()

rolling_origin(NYSE_other %>% arrange(date), initial = 30, assess = 7) %>%
#sliding_period(NYSE_other %>% arrange(date), date, period = "day", lookback = Inf, assess_stop = 1) %>% 
  mutate(train_data = map(splits, analysis),
         test_data = map(splits, assessment)) %>% 
  select(-splits) %>% 
  pivot_longer(-id) %>% 
  filter(id %in% c("Slice0001", "Slice0002", "Slice0003")) %>% 
  unnest(value) %>% 
  ggplot(aes(x = date, y = log_volume, color = name, group = NA)) + 
  geom_point() + 
  geom_line() +
  facet_wrap(~id, scales = "fixed")

sliding_period(NYSE_other %>% arrange(date), 
               date, period = "month", lookback = Inf, assess_stop = 1) %>% 
  mutate(train_data = map(splits, analysis),
         test_data = map(splits, assessment)) %>% 
  select(-splits) %>% 
  pivot_longer(-id) %>% 
  filter(id %in% c("Slice001", "Slice002", "Slice003")) %>% 
  unnest(value) %>% 
  ggplot(aes(x = date, y = log_volume, color = name, group = NA)) + 
  geom_point() +
  geom_line() + 
  facet_wrap(~id, scales = "fixed")

  • Rolling forecast origin resampling (Hyndman and Athanasopoulos 2018) provides a method that emulates how time series data is often partitioned in practice, estimating the model with historical data and evaluating it with the most recent data.

  • Tune AR(5) with elastic net (lasso + ridge) regularization using all 3 features on the training data, and evaluate the test performance.

1.4 Preprocessing

en_receipe <- 
  recipe(log_volume ~ ., data = NYSE_other) %>% 
  step_dummy(all_nominal(), -all_outcomes()) %>% 
  step_normalize(all_numeric_predictors(), -all_outcomes()) %>%  
  step_naomit(all_predictors()) %>%
  prep(data = NYSE_other)
en_receipe

1.5 Model training

### Model
enet_mod <- 
  # mixture = 0 (ridge), mixture = 1 (lasso)
  # mixture = (0, 1) elastic net 
  # As an example, we set mixture = 0.5. It needs to be tuned.
  linear_reg(penalty = tune(), mixture = 0.5) %>% 
  set_engine("glmnet")
enet_mod
Linear Regression Model Specification (regression)

Main Arguments:
  penalty = tune()
  mixture = 0.5

Computational engine: glmnet 
en_wf <- 
  workflow() %>%
  add_model(enet_mod) %>%
  add_recipe(en_receipe %>% step_rm(date) %>% step_indicate_na())
en_wf
══ Workflow ════════════════════════════════════════════════════════════════════
Preprocessor: Recipe
Model: linear_reg()

── Preprocessor ────────────────────────────────────────────────────────────────
5 Recipe Steps

• step_dummy()
• step_normalize()
• step_naomit()
• step_rm()
• step_indicate_na()

── Model ───────────────────────────────────────────────────────────────────────
Linear Regression Model Specification (regression)

Main Arguments:
  penalty = tune()
  mixture = 0.5

Computational engine: glmnet 
folds <- NYSE_other %>% arrange(date) %>%
    sliding_period(date, period = "month", lookback = Inf, assess_stop = 1)
  # rolling_origin(initial = 5, assess = 1)
folds
# Sliding period resampling 
# A tibble: 204 × 2
   splits           id      
   <list>           <chr>   
 1 <split [15/22]>  Slice001
 2 <split [37/19]>  Slice002
 3 <split [56/21]>  Slice003
 4 <split [77/21]>  Slice004
 5 <split [98/22]>  Slice005
 6 <split [120/20]> Slice006
 7 <split [140/22]> Slice007
 8 <split [162/22]> Slice008
 9 <split [184/20]> Slice009
10 <split [204/23]> Slice010
# ℹ 194 more rows
month_folds <- NYSE_other %>%
  sliding_period(
    date,
    "month",
    lookback = Inf,
    skip = 4)
month_folds
# Sliding period resampling 
# A tibble: 200 × 2
   splits           id      
   <list>           <chr>   
 1 <split [98/22]>  Slice001
 2 <split [120/20]> Slice002
 3 <split [140/22]> Slice003
 4 <split [162/22]> Slice004
 5 <split [184/20]> Slice005
 6 <split [204/23]> Slice006
 7 <split [227/18]> Slice007
 8 <split [245/21]> Slice008
 9 <split [266/22]> Slice009
10 <split [288/19]> Slice010
# ℹ 190 more rows
lambda_grid <-
  grid_regular(penalty(range = c(-8, -7), trans = log10_trans()), levels = 3)
lambda_grid
# A tibble: 3 × 1
       penalty
         <dbl>
1 0.00000001  
2 0.0000000316
3 0.0000001   
en_fit <- tune_grid(en_wf, resamples = month_folds, grid = lambda_grid) 
en_fit
# Tuning results
# Sliding period resampling 
# A tibble: 200 × 4
   splits           id       .metrics         .notes          
   <list>           <chr>    <list>           <list>          
 1 <split [98/22]>  Slice001 <tibble [6 × 5]> <tibble [0 × 3]>
 2 <split [120/20]> Slice002 <tibble [6 × 5]> <tibble [0 × 3]>
 3 <split [140/22]> Slice003 <tibble [6 × 5]> <tibble [0 × 3]>
 4 <split [162/22]> Slice004 <tibble [6 × 5]> <tibble [0 × 3]>
 5 <split [184/20]> Slice005 <tibble [6 × 5]> <tibble [0 × 3]>
 6 <split [204/23]> Slice006 <tibble [6 × 5]> <tibble [0 × 3]>
 7 <split [227/18]> Slice007 <tibble [6 × 5]> <tibble [0 × 3]>
 8 <split [245/21]> Slice008 <tibble [6 × 5]> <tibble [0 × 3]>
 9 <split [266/22]> Slice009 <tibble [6 × 5]> <tibble [0 × 3]>
10 <split [288/19]> Slice010 <tibble [6 × 5]> <tibble [0 × 3]>
# ℹ 190 more rows
en_fit %>%
  collect_metrics() %>%
  print(width = Inf) %>%
  filter(.metric == "rsq") %>%
  ggplot(mapping = aes(x = penalty, y = mean)) + 
  geom_point() + 
  geom_line() + 
  labs(x = "Penalty", y = "CV RMSE") + 
  scale_x_log10(labels = scales::label_number())
# A tibble: 6 × 7
       penalty .metric .estimator  mean     n std_err .config             
         <dbl> <chr>   <chr>      <dbl> <int>   <dbl> <chr>               
1 0.00000001   rmse    standard   0.133   200 0.00287 Preprocessor1_Model1
2 0.00000001   rsq     standard   0.381   200 0.0149  Preprocessor1_Model1
3 0.0000000316 rmse    standard   0.133   200 0.00287 Preprocessor1_Model2
4 0.0000000316 rsq     standard   0.381   200 0.0149  Preprocessor1_Model2
5 0.0000001    rmse    standard   0.133   200 0.00287 Preprocessor1_Model3
6 0.0000001    rsq     standard   0.381   200 0.0149  Preprocessor1_Model3

en_fit %>%
  show_best("rsq")
# A tibble: 3 × 7
       penalty .metric .estimator  mean     n std_err .config             
         <dbl> <chr>   <chr>      <dbl> <int>   <dbl> <chr>               
1 0.00000001   rsq     standard   0.381   200  0.0149 Preprocessor1_Model1
2 0.0000000316 rsq     standard   0.381   200  0.0149 Preprocessor1_Model2
3 0.0000001    rsq     standard   0.381   200  0.0149 Preprocessor1_Model3
best_en_fit <- en_fit %>%
  select_best("rsq")
best_en_fit
# A tibble: 1 × 2
     penalty .config             
       <dbl> <chr>               
1 0.00000001 Preprocessor1_Model1
# Final workflow
final_wf_en <- en_wf %>%
  finalize_workflow(best_en_fit)
final_wf_en
══ Workflow ════════════════════════════════════════════════════════════════════
Preprocessor: Recipe
Model: linear_reg()

── Preprocessor ────────────────────────────────────────────────────────────────
5 Recipe Steps

• step_dummy()
• step_normalize()
• step_naomit()
• step_rm()
• step_indicate_na()

── Model ───────────────────────────────────────────────────────────────────────
Linear Regression Model Specification (regression)

Main Arguments:
  penalty = 1e-08
  mixture = 0.5

Computational engine: glmnet 
# Fit the whole training set, then predict the test cases
final_fit_en <- 
  final_wf_en %>%
  last_fit(correct_split)
final_fit_en
# Resampling results
# Manual resampling 
# A tibble: 1 × 6
  splits              id               .metrics .notes   .predictions .workflow 
  <list>              <chr>            <list>   <list>   <list>       <list>    
1 <split [3207/1069]> train/test split <tibble> <tibble> <tibble>     <workflow>
# Test metrics
final_fit_en %>% collect_metrics()
# A tibble: 2 × 4
  .metric .estimator .estimate .config             
  <chr>   <chr>          <dbl> <chr>               
1 rmse    standard       0.138 Preprocessor1_Model1
2 rsq     standard       0.688 Preprocessor1_Model1
predictions <- final_fit_en %>%
  collect_predictions()
predictions
# A tibble: 1,069 × 5
   id                 .pred  .row log_volume .config             
   <chr>              <dbl> <int>      <dbl> <chr>               
 1 train/test split -0.135   3208     0.0353 Preprocessor1_Model1
 2 train/test split -0.0388  3209     0.0338 Preprocessor1_Model1
 3 train/test split -0.106   3210    -0.141  Preprocessor1_Model1
 4 train/test split -0.0471  3211    -0.352  Preprocessor1_Model1
 5 train/test split -0.160   3212     0.154  Preprocessor1_Model1
 6 train/test split  0.0224  3213    -0.167  Preprocessor1_Model1
 7 train/test split -0.116   3214     0.102  Preprocessor1_Model1
 8 train/test split -0.0778  3215    -0.0838 Preprocessor1_Model1
 9 train/test split -0.0669  3216    -0.247  Preprocessor1_Model1
10 train/test split -0.0560  3217     0.205  Preprocessor1_Model1
# ℹ 1,059 more rows

1.6 Random forest forecaster (30pts)

  • Use the same features as in AR(\(L\)) for the random forest. Tune the random forest and evaluate the test performance.

  • Hint: Workflow: Random Forest for Prediction is a good starting point.

rf_recipe <- recipe(log_volume ~ ., data = NYSE_other) %>% 
  step_dummy(all_nominal(), -all_outcomes()) %>% 
  step_normalize(all_numeric_predictors(), -all_outcomes()) %>%  
  step_naomit(all_predictors()) %>%
  prep(data = NYSE_other)
rf_recipe
rf_mod <- 
  rand_forest(
    mode = "regression",
    # Number of predictors randomly sampled in each split
    mtry = tune(),
    # Number of trees in ensemble
    trees = tune()
  ) %>% 
  set_engine("ranger")
rf_mod
Random Forest Model Specification (regression)

Main Arguments:
  mtry = tune()
  trees = tune()

Computational engine: ranger 
rf_wf <- workflow() %>%
   add_recipe(rf_recipe %>% step_rm(date) %>% step_indicate_na()) %>%
  add_model(rf_mod)
rf_wf
══ Workflow ════════════════════════════════════════════════════════════════════
Preprocessor: Recipe
Model: rand_forest()

── Preprocessor ────────────────────────────────────────────────────────────────
5 Recipe Steps

• step_dummy()
• step_normalize()
• step_naomit()
• step_rm()
• step_indicate_na()

── Model ───────────────────────────────────────────────────────────────────────
Random Forest Model Specification (regression)

Main Arguments:
  mtry = tune()
  trees = tune()

Computational engine: ranger 
folds <- NYSE_other %>% arrange(date) %>%
    sliding_period(date, period = "month", lookback = Inf, assess_stop = 1)
  # rolling_origin(initial = 5, assess = 1)
folds
# Sliding period resampling 
# A tibble: 204 × 2
   splits           id      
   <list>           <chr>   
 1 <split [15/22]>  Slice001
 2 <split [37/19]>  Slice002
 3 <split [56/21]>  Slice003
 4 <split [77/21]>  Slice004
 5 <split [98/22]>  Slice005
 6 <split [120/20]> Slice006
 7 <split [140/22]> Slice007
 8 <split [162/22]> Slice008
 9 <split [184/20]> Slice009
10 <split [204/23]> Slice010
# ℹ 194 more rows
month_folds <- NYSE_other %>%
  sliding_period(
    date,
    "month",
    lookback = Inf,
    skip = 4)
month_folds
# Sliding period resampling 
# A tibble: 200 × 2
   splits           id      
   <list>           <chr>   
 1 <split [98/22]>  Slice001
 2 <split [120/20]> Slice002
 3 <split [140/22]> Slice003
 4 <split [162/22]> Slice004
 5 <split [184/20]> Slice005
 6 <split [204/23]> Slice006
 7 <split [227/18]> Slice007
 8 <split [245/21]> Slice008
 9 <split [266/22]> Slice009
10 <split [288/19]> Slice010
# ℹ 190 more rows
rf_grid <- grid_regular(
  trees(range = c(100L, 300L)), 
  mtry(range = c(1L, 5L)),
  levels = c(3, 5)
  )
rf_grid
# A tibble: 15 × 2
   trees  mtry
   <int> <int>
 1   100     1
 2   200     1
 3   300     1
 4   100     2
 5   200     2
 6   300     2
 7   100     3
 8   200     3
 9   300     3
10   100     4
11   200     4
12   300     4
13   100     5
14   200     5
15   300     5
rf_fit <- tune_grid(
  rf_wf,
  resamples = month_folds,
  grid = rf_grid,
 metrics = metric_set(rmse, rsq)  
)
rf_fit
# Tuning results
# Sliding period resampling 
# A tibble: 200 × 4
   splits           id       .metrics          .notes          
   <list>           <chr>    <list>            <list>          
 1 <split [98/22]>  Slice001 <tibble [30 × 6]> <tibble [0 × 3]>
 2 <split [120/20]> Slice002 <tibble [30 × 6]> <tibble [0 × 3]>
 3 <split [140/22]> Slice003 <tibble [30 × 6]> <tibble [0 × 3]>
 4 <split [162/22]> Slice004 <tibble [30 × 6]> <tibble [0 × 3]>
 5 <split [184/20]> Slice005 <tibble [30 × 6]> <tibble [0 × 3]>
 6 <split [204/23]> Slice006 <tibble [30 × 6]> <tibble [0 × 3]>
 7 <split [227/18]> Slice007 <tibble [30 × 6]> <tibble [0 × 3]>
 8 <split [245/21]> Slice008 <tibble [30 × 6]> <tibble [0 × 3]>
 9 <split [266/22]> Slice009 <tibble [30 × 6]> <tibble [0 × 3]>
10 <split [288/19]> Slice010 <tibble [30 × 6]> <tibble [0 × 3]>
# ℹ 190 more rows
rf_fit %>%
  collect_metrics() %>%
  print(width = Inf) %>%
  filter(.metric == "rsq") %>%
  mutate(mtry = as.factor(mtry)) %>%
  ggplot(mapping = aes(x = trees, y = mean, color = mtry)) +
  # geom_point() + 
  geom_line() + 
  labs(x = "Num. of Trees", y = "CV mse")
# A tibble: 30 × 8
    mtry trees .metric .estimator  mean     n std_err .config              
   <int> <int> <chr>   <chr>      <dbl> <int>   <dbl> <chr>                
 1     1   100 rmse    standard   0.161   200 0.00391 Preprocessor1_Model01
 2     1   100 rsq     standard   0.306   200 0.0138  Preprocessor1_Model01
 3     1   200 rmse    standard   0.161   200 0.00389 Preprocessor1_Model02
 4     1   200 rsq     standard   0.311   200 0.0143  Preprocessor1_Model02
 5     1   300 rmse    standard   0.161   200 0.00389 Preprocessor1_Model03
 6     1   300 rsq     standard   0.316   200 0.0141  Preprocessor1_Model03
 7     2   100 rmse    standard   0.147   200 0.00333 Preprocessor1_Model04
 8     2   100 rsq     standard   0.320   200 0.0143  Preprocessor1_Model04
 9     2   200 rmse    standard   0.147   200 0.00330 Preprocessor1_Model05
10     2   200 rsq     standard   0.330   200 0.0145  Preprocessor1_Model05
# ℹ 20 more rows

# Final workflow
rf_fit %>%
  show_best("rsq")
# A tibble: 5 × 8
   mtry trees .metric .estimator  mean     n std_err .config              
  <int> <int> <chr>   <chr>      <dbl> <int>   <dbl> <chr>                
1     5   300 rsq     standard   0.342   200  0.0148 Preprocessor1_Model15
2     4   200 rsq     standard   0.339   200  0.0147 Preprocessor1_Model11
3     5   200 rsq     standard   0.338   200  0.0146 Preprocessor1_Model14
4     3   200 rsq     standard   0.338   200  0.0146 Preprocessor1_Model08
5     5   100 rsq     standard   0.337   200  0.0145 Preprocessor1_Model13
best_rf <- rf_fit %>%
  select_best("rsq")
best_rf
# A tibble: 1 × 3
   mtry trees .config              
  <int> <int> <chr>                
1     5   300 Preprocessor1_Model15
# Final workflow
final_wf_rf <- rf_wf %>%
  finalize_workflow(best_rf)
final_wf_rf
══ Workflow ════════════════════════════════════════════════════════════════════
Preprocessor: Recipe
Model: rand_forest()

── Preprocessor ────────────────────────────────────────────────────────────────
5 Recipe Steps

• step_dummy()
• step_normalize()
• step_naomit()
• step_rm()
• step_indicate_na()

── Model ───────────────────────────────────────────────────────────────────────
Random Forest Model Specification (regression)

Main Arguments:
  mtry = 5
  trees = 300

Computational engine: ranger 
# Fit the whole training set, then predict the test cases
final_fit_rf <- 
  final_wf_rf %>%
  last_fit(correct_split)
final_fit_rf
# Resampling results
# Manual resampling 
# A tibble: 1 × 6
  splits              id               .metrics .notes   .predictions .workflow 
  <list>              <chr>            <list>   <list>   <list>       <list>    
1 <split [3207/1069]> train/test split <tibble> <tibble> <tibble>     <workflow>
# Test metrics
final_fit_rf %>% 
  collect_metrics()
# A tibble: 2 × 4
  .metric .estimator .estimate .config             
  <chr>   <chr>          <dbl> <chr>               
1 rmse    standard       0.151 Preprocessor1_Model1
2 rsq     standard       0.652 Preprocessor1_Model1

1.7 Boosting forecaster (30pts)

  • Use the same features as in AR(\(L\)) for the boosting. Tune the boosting algorithm and evaluate the test performance.

  • Hint: Workflow: Boosting tree for Prediction is a good starting point.

boost_recipe <- recipe(log_volume ~ ., data = NYSE_other) %>% 
  step_dummy(all_nominal(), -all_outcomes()) %>% 
  step_normalize(all_numeric_predictors(), -all_outcomes()) %>%  
  step_naomit(all_predictors()) %>%
  prep(data = NYSE_other)
boost_recipe
gb_mod <- 
  boost_tree(
    mode = "regression",
    trees = 1000, 
    tree_depth = tune(),
    learn_rate = tune()
  ) %>% 
  set_engine("xgboost")
gb_mod
Boosted Tree Model Specification (regression)

Main Arguments:
  trees = 1000
  tree_depth = tune()
  learn_rate = tune()

Computational engine: xgboost 
boost_workflow <- workflow() %>%
  add_model(gb_mod) %>%
  add_recipe(boost_recipe %>% step_rm(date) %>% step_indicate_na())
boost_workflow
══ Workflow ════════════════════════════════════════════════════════════════════
Preprocessor: Recipe
Model: boost_tree()

── Preprocessor ────────────────────────────────────────────────────────────────
5 Recipe Steps

• step_dummy()
• step_normalize()
• step_naomit()
• step_rm()
• step_indicate_na()

── Model ───────────────────────────────────────────────────────────────────────
Boosted Tree Model Specification (regression)

Main Arguments:
  trees = 1000
  tree_depth = tune()
  learn_rate = tune()

Computational engine: xgboost 
folds <- NYSE_other %>% arrange(date) %>%
    sliding_period(date, period = "month", lookback = Inf, assess_stop = 1)
  # rolling_origin(initial = 5, assess = 1)


month_folds <- NYSE_other %>%
  sliding_period(
    date,
    "month",
    lookback = Inf,
    skip = 4)
boost_grid <- grid_regular(
  tree_depth(range = c(1L, 2L)), 
  learn_rate(range = c(-3, -2), trans = log10_trans()), 
  levels = c(2, 3) 
)
boost_grid
# A tibble: 6 × 2
  tree_depth learn_rate
       <int>      <dbl>
1          1    0.001  
2          2    0.001  
3          1    0.00316
4          2    0.00316
5          1    0.01   
6          2    0.01   
boost_fit <- tune_grid(
  boost_workflow,
  resamples = month_folds,
  grid = boost_grid,
  metrics = metric_set(rmse, rsq)
)
boost_fit
# Tuning results
# Sliding period resampling 
# A tibble: 200 × 4
   splits           id       .metrics          .notes          
   <list>           <chr>    <list>            <list>          
 1 <split [98/22]>  Slice001 <tibble [12 × 6]> <tibble [0 × 3]>
 2 <split [120/20]> Slice002 <tibble [12 × 6]> <tibble [0 × 3]>
 3 <split [140/22]> Slice003 <tibble [12 × 6]> <tibble [0 × 3]>
 4 <split [162/22]> Slice004 <tibble [12 × 6]> <tibble [0 × 3]>
 5 <split [184/20]> Slice005 <tibble [12 × 6]> <tibble [0 × 3]>
 6 <split [204/23]> Slice006 <tibble [12 × 6]> <tibble [0 × 3]>
 7 <split [227/18]> Slice007 <tibble [12 × 6]> <tibble [0 × 3]>
 8 <split [245/21]> Slice008 <tibble [12 × 6]> <tibble [0 × 3]>
 9 <split [266/22]> Slice009 <tibble [12 × 6]> <tibble [0 × 3]>
10 <split [288/19]> Slice010 <tibble [12 × 6]> <tibble [0 × 3]>
# ℹ 190 more rows
boost_fit %>%
  collect_metrics() %>%
  print(width = Inf) %>%
  filter(.metric == "rsq") %>%
  ggplot(mapping = aes(x = learn_rate, y = mean, color = factor(tree_depth))) +
  geom_point() +
  geom_line() +
  labs(x = "Learning Rate", y = "CV AUC") +
  scale_x_log10()
# A tibble: 12 × 8
   tree_depth learn_rate .metric .estimator  mean     n std_err
        <int>      <dbl> <chr>   <chr>      <dbl> <int>   <dbl>
 1          1    0.001   rmse    standard   0.257   200 0.00562
 2          1    0.001   rsq     standard   0.203   200 0.0120 
 3          2    0.001   rmse    standard   0.251   200 0.00502
 4          2    0.001   rsq     standard   0.239   200 0.0126 
 5          1    0.00316 rmse    standard   0.160   200 0.00356
 6          1    0.00316 rsq     standard   0.253   200 0.0129 
 7          2    0.00316 rmse    standard   0.150   200 0.00313
 8          2    0.00316 rsq     standard   0.303   200 0.0141 
 9          1    0.01    rmse    standard   0.144   200 0.00307
10          1    0.01    rsq     standard   0.322   200 0.0143 
11          2    0.01    rmse    standard   0.136   200 0.00283
12          2    0.01    rsq     standard   0.369   200 0.0153 
   .config             
   <chr>               
 1 Preprocessor1_Model1
 2 Preprocessor1_Model1
 3 Preprocessor1_Model2
 4 Preprocessor1_Model2
 5 Preprocessor1_Model3
 6 Preprocessor1_Model3
 7 Preprocessor1_Model4
 8 Preprocessor1_Model4
 9 Preprocessor1_Model5
10 Preprocessor1_Model5
11 Preprocessor1_Model6
12 Preprocessor1_Model6

boost_fit %>%
  show_best("rsq")
# A tibble: 5 × 8
  tree_depth learn_rate .metric .estimator  mean     n std_err .config          
       <int>      <dbl> <chr>   <chr>      <dbl> <int>   <dbl> <chr>            
1          2    0.01    rsq     standard   0.369   200  0.0153 Preprocessor1_Mo…
2          1    0.01    rsq     standard   0.322   200  0.0143 Preprocessor1_Mo…
3          2    0.00316 rsq     standard   0.303   200  0.0141 Preprocessor1_Mo…
4          1    0.00316 rsq     standard   0.253   200  0.0129 Preprocessor1_Mo…
5          2    0.001   rsq     standard   0.239   200  0.0126 Preprocessor1_Mo…
best_gb <- boost_fit %>%
  select_best("rsq")
best_gb
# A tibble: 1 × 3
  tree_depth learn_rate .config             
       <int>      <dbl> <chr>               
1          2       0.01 Preprocessor1_Model6
# Final workflow
final_wf_boost <- boost_workflow %>%
  finalize_workflow(best_gb)
final_wf_boost
══ Workflow ════════════════════════════════════════════════════════════════════
Preprocessor: Recipe
Model: boost_tree()

── Preprocessor ────────────────────────────────────────────────────────────────
5 Recipe Steps

• step_dummy()
• step_normalize()
• step_naomit()
• step_rm()
• step_indicate_na()

── Model ───────────────────────────────────────────────────────────────────────
Boosted Tree Model Specification (regression)

Main Arguments:
  trees = 1000
  tree_depth = 2
  learn_rate = 0.01

Computational engine: xgboost 
# Fit the whole training set, then predict the test cases
final_fit_boost <- 
  final_wf_boost %>%
  last_fit(correct_split)
final_fit_boost
# Resampling results
# Manual resampling 
# A tibble: 1 × 6
  splits              id               .metrics .notes   .predictions .workflow 
  <list>              <chr>            <list>   <list>   <list>       <list>    
1 <split [3207/1069]> train/test split <tibble> <tibble> <tibble>     <workflow>
# Test metrics
final_fit_boost %>% 
  collect_metrics()
# A tibble: 2 × 4
  .metric .estimator .estimate .config             
  <chr>   <chr>          <dbl> <chr>               
1 rmse    standard       0.147 Preprocessor1_Model1
2 rsq     standard       0.665 Preprocessor1_Model1

1.8 Summary (30pts)

Your score for this question is largely determined by your final test performance.

Summarize the performance of different machine learning forecasters in the following format.

The summary of the performance of various machine learning forecasters on predicting the daily Log trading volume using the New York Stock Exchange data from 1962 to 1986, with training data up to January 2, 1980, is presented below. The evaluation of model performance was based on the coefficient of determination (R^2), both on cross-validation (CV) and on the unseen test data. The R^2 metric measures the proportion of variance in the dependent variable that is predictable from the independent variables, with values closer to 1 indicating better model performance. | Method | CV \(R^2\) | Test \(R^2\) | |:——:|:——:|:——:|:——:| | Baseline | - |Straw man test R2: 0.35 | | AR(5) | 0.38 | 0.69 | | Random Forest | 0.34| 0.65| | Boosting | 0.37| 0.67| Baseline (Straw man): The baseline model, which uses the previous day’s log trading volume as the prediction for the current day, achieved a R^2 of 0.35 on the test set. This simple model serves as a point of comparison for more complex models.

AR(5): An autoregressive model of order 5 (AR(5)), which uses the past 5 days of log trading volume to predict the current day’s volume, showed an improvement over the baseline, with a CV R^2 of 0.38 and a test R^2 of 0.69. This indicates that incorporating more historical data points helps in improving the prediction accuracy.

Random Forest: The Random Forest model, which is a non-linear and more complex model compared to AR(5), achieved a slightly lower CV R^2 of 0.34 but also a lower test R^2 of 0.65. This suggests that despite its ability to capture complex patterns in the data, it might be overfitting to the training data or not capturing the time series dynamics as effectively as linear models in this specific case.

Boosting: The Boosting model, another sophisticated and flexible ensemble method, showed a CV R^2 of 0.37 and a test R^2 of 0.67. It performed better than the Random Forest model but was still slightly behind the AR(5) model in terms of test performance. Boosting’s iterative approach to correcting errors might be beneficial for this time series forecasting task, although it didn’t outperform the simpler AR(5) model.

Conclusion: The AR(5) model, despite its simplicity, provided the best performance on the test data among the models evaluated. This suggests that for the task of forecasting daily Log trading volume, a linear approach that directly incorporates recent historical values can be very effective. Meanwhile, the more complex Random Forest and Boosting models did not achieve higher performance, highlighting the importance of model selection and the possibility that simpler models may suffice for certain forecasting tasks. ### Extension reading
- MOIRAI: Salesforce’s Foundation Model for Time-Series Forecasting

2 ISL Exercise 12.6.13 (90 pts)

  1. On the book website, www.statlearning.com, there is a gene expression data set (Ch12Ex13.csv) that consists of 40 tissue samples with measurements on 1,000 genes. The first 20 samples are from healthy patients, while the second 20 are from a diseased group. Load in the data using read.csv(). You will need to select header = F.
  1. Apply hierarchical clustering to the samples using correlation- based distance, and plot the dendrogram. Do the genes separate the samples into the two groups? Do your results depend on the type of linkage used? PCA and UMAP
  2. Your collaborator wants to know which genes differ the most across the two groups. Suggest a way to answer this question, and apply it here.

import data

library(tidyverse)
library(cluster)
library(ggplot2)
# Load the data
gene_express <- read.csv('../hw6/Ch12Ex13.csv', header = FALSE, col.names = paste("ID", 1:40, sep = ""))

# Few top rows of the dataset:
head(gene_express)
          ID1        ID2        ID3        ID4        ID5        ID6
1 -0.96193340  0.4418028 -0.9750051  1.4175040  0.8188148  0.3162937
2 -0.29252570 -1.1392670  0.1958370 -1.2811210 -0.2514393  2.5119970
3  0.25878820 -0.9728448  0.5884858 -0.8002581 -1.8203980 -2.0589240
4 -1.15213200 -2.2131680 -0.8615249  0.6309253  0.9517719 -1.1657240
5  0.19578280  0.5933059  0.2829921  0.2471472  1.9786680 -0.8710180
6  0.03012394 -0.6910143 -0.4034258 -0.7298590 -0.3640986  1.1253490
          ID7         ID8         ID9       ID10       ID11       ID12
1 -0.02496682 -0.06396600  0.03149702 -0.3503106 -0.7227299 -0.2819547
2 -0.92220620  0.05954277 -1.40964500 -0.6567122 -0.1157652  0.8259783
3 -0.06476437  1.59212400 -0.17311700 -0.1210874 -0.1875790 -1.5001630
4 -0.39155860  1.06361900 -0.35000900 -1.4890580 -0.2432189 -0.4330340
5 -0.98971500 -1.03225300 -1.10965400 -0.3851423  1.6509570 -1.7449090
6 -1.40404100 -0.80613040 -1.23792400  0.5776018 -0.2720642  2.1765620
         ID13        ID14       ID15       ID16       ID17       ID18
1  1.33751500  0.70197980  1.0076160 -0.4653828  0.6385951  0.2867807
2  0.34644960 -0.56954860 -0.1315365  0.6902290 -0.9090382  1.3026420
3 -1.22873700  0.85598900  1.2498550 -0.8980815  0.8702058 -0.2252529
4 -0.03879128 -0.05789677 -1.3977620 -0.1561871 -2.7359820  0.7756169
5 -0.37888530 -0.67982610 -2.1315840 -0.2301718  0.4661243 -1.8004490
6  1.43640700 -1.02578100  0.2981582 -0.5559659  0.2046529 -1.1916480
        ID19        ID20       ID21       ID22       ID23       ID24
1 -0.2270782 -0.22004520 -1.2425730 -0.1085056 -1.8642620 -0.5005122
2 -1.6726950 -0.52550400  0.7979700 -0.6897930  0.8995305  0.4285812
3  0.4502892  0.55144040  0.1462943  0.1297400  1.3042290 -1.6619080
4  0.6141562  2.01919400  1.0811390 -1.0766180 -0.2434181  0.5134822
5  0.6262904 -0.09772305 -0.2997108 -0.5295591 -2.0235670 -0.5108402
6  0.2350916  0.67096470  0.1307988  1.0689940  1.2309870  1.1344690
         ID25        ID26       ID27       ID28        ID29        ID30
1 -1.32500800  1.06341100 -0.2963712 -0.1216457  0.08516605  0.62417640
2 -0.67611410 -0.53409490 -1.7325070 -1.6034470 -1.08362000  0.03342185
3 -1.63037600 -0.07742528  1.3061820  0.7926002  1.55946500 -0.68851160
4 -0.51285780  2.55167600 -2.3143010 -1.2764700 -1.22927100  1.43439600
5  0.04600274  1.26803000 -0.7439868  0.2231319  0.85846280  0.27472610
6  0.55636800 -0.35876640  1.0798650 -0.2064905 -0.00616453  0.16425470
        ID31         ID32        ID33       ID34       ID35       ID36
1 -0.5095915 -0.216725500 -0.05550597 -0.4844491 -0.5215811  1.9491350
2  1.7007080  0.007289556  0.09906234  0.5638533 -0.2572752 -0.5817805
3 -0.6154720  0.009999363  0.94581000 -0.3185212 -0.1178895  0.6213662
4 -0.2842774  0.198945600 -0.09183320  0.3496279 -0.2989097  1.5136960
5 -0.6929984 -0.845707200 -0.17749680 -0.1664908  1.4831550 -1.6879460
6  1.1567370  0.241774500  0.08863952  0.1829540  0.9426771 -0.2096004
         ID37       ID38        ID39       ID40
1  1.32433500  0.4681471  1.06110000  1.6559700
2 -0.16988710 -0.5423036  0.31293890 -1.2843770
3 -0.07076396  0.4016818 -0.01622713 -0.5265532
4  0.67118470  0.0108553 -1.04368900  1.6252750
5 -0.14142960  0.2007785 -0.67594210  2.2206110
6  0.53626210 -1.1852260 -0.42274760  0.6243603
grp = factor(rep(c(1, 0), each = 20))

regression <- function(y) {
  sum <- summary(lm(y ~ grp))
  pv <- sum$coefficients[2, 4]
  return(pv)
}


out <- tibble(gene = seq(1, nrow(gene_express)),
              p_values = unlist(purrr:: map(1:nrow(gene_express), ~regression(as.matrix(gene_express)[.x, ]))))
out %>% arrange(p_values) %>% head(10)
# A tibble: 10 × 2
    gene p_values
   <int>    <dbl>
 1   502 1.43e-12
 2   589 3.19e-12
 3   600 9.81e-12
 4   590 6.28e-11
 5   565 1.74e-10
 6   551 1.10e- 9
 7   593 1.19e- 9
 8   584 1.65e- 9
 9   538 2.42e- 9
10   569 2.69e- 9
# sig <- out %>% arrange(p_values) %>% filter(p_values < 0.05/nrow(Ch12Ex13))
sig <- out %>% arrange(p_values) %>% filter(p_values < 0.05 )
# install.packages("pheatmap")
library(pheatmap)
# install.packages("ggplotify")
library(ggplotify) ## to convert pheatmap to ggplot2
# install.packages("heatmaply")
library(heatmaply) ## for constructing interactive heatmap
#create data frame for annotations
dfh <- data.frame(sample=as.character(colnames(gene_express)), status = "disease") %>%
                column_to_rownames("sample")
dfh$status[seq(21, 40)] <-  "healthy"
dfh
      status
ID1  disease
ID2  disease
ID3  disease
ID4  disease
ID5  disease
ID6  disease
ID7  disease
ID8  disease
ID9  disease
ID10 disease
ID11 disease
ID12 disease
ID13 disease
ID14 disease
ID15 disease
ID16 disease
ID17 disease
ID18 disease
ID19 disease
ID20 disease
ID21 healthy
ID22 healthy
ID23 healthy
ID24 healthy
ID25 healthy
ID26 healthy
ID27 healthy
ID28 healthy
ID29 healthy
ID30 healthy
ID31 healthy
ID32 healthy
ID33 healthy
ID34 healthy
ID35 healthy
ID36 healthy
ID37 healthy
ID38 healthy
ID39 healthy
ID40 healthy
pheatmap(gene_express[sig$gene, ], cluster_rows = FALSE, cluster_cols = T, scale="row", annotation_col = dfh,
         annotation_colors=list(status = c(disease = "orange", healthy = "black")),
         color=colorRampPalette(c("navy", "white", "red"))(50))

# Transpose 'gene_express' to have samples as rows and genes as columns
gene_express_t <- t(gene_express)

# Convert to data frame and add 'status' information
gene_express_df <- as.data.frame(gene_express_t) %>%
  mutate(status = dfh$status)

2.1 12.6.13 (b) (30 pts)

There are several methods of “dissimilarity” (correlation based distance) which can be explored for creation of distance matrix. We now plot the hierarchical clustering dendrogram using complete, single, ward centroid and average linkage clustering using correlation based distance. single linkage:

library(tidyverse)
library(tidymodels)
library(readr)
library(tswge)
library(ggplot2)
library(yardstick)
library(workflows)
library(parsnip)
library(tidyclust)
library(RcppHungarian)
library(embed)
library(tidytext)
set.seed(838383)
hc_spec_single <- hier_clust(
   num_clusters = 2,
  linkage_method = "single"
)


hc_fit_single <- hc_spec_single %>%
  fit(~ .,
    data = gene_express_df 
  )

hc_fit_single %>%
  summary()
        Length Class      Mode
spec    4      hier_clust list
fit     7      hclust     list
elapsed 1      -none-     list
preproc 4      -none-     list
hc_fit_single$fit %>% plot(main=" Single Linkage
with Correlation - Based Distance ", xlab="", sub ="")

average linkage:

set.seed(838383)
hc_spec_average <- hier_clust(
   num_clusters = 2,
  linkage_method = "average"
)


hc_fit_average <- hc_spec_average %>%
  fit(~ .,
    data = gene_express_df 
  )

hc_fit_average %>%
  summary()
        Length Class      Mode
spec    4      hier_clust list
fit     7      hclust     list
elapsed 1      -none-     list
preproc 4      -none-     list
hc_fit_average$fit %>% plot(main=" Average Linkage
with Correlation - Based Distance ", xlab="", sub ="")

complete linkage:

set.seed(838383)
hc_spec_complete <- hier_clust(
   num_clusters = 2,
  linkage_method = "complete"
)


hc_fit_complete <- hc_spec_complete %>%
  fit(~ .,
    data = gene_express_df
  )

hc_fit_complete %>%
  summary()
        Length Class      Mode
spec    4      hier_clust list
fit     7      hclust     list
elapsed 1      -none-     list
preproc 4      -none-     list
hc_fit_complete$fit %>% plot(main=" Complete Linkage
with Correlation - Based Distance ", xlab="", sub ="")

centroid linkage:

set.seed(838383)
hc_spec_centroid <- hier_clust(
   num_clusters = 2,
  linkage_method = "centroid"
)
hc_fit_centroid <- hc_spec_centroid %>%
  fit(~ .,
    data = gene_express_df
  )

hc_fit_centroid %>%
  summary()
        Length Class      Mode
spec    4      hier_clust list
fit     7      hclust     list
elapsed 1      -none-     list
preproc 4      -none-     list
hc_fit_centroid$fit %>% plot(main=" Centroid Linkage
with Correlation - Based Distance ", xlab="", sub ="")

ward linkage:

set.seed(838383)
hc_spec_ward <- hier_clust(
   num_clusters = 2,
  linkage_method = "ward.D"
)
hc_fit_ward<- hc_spec_ward %>%
  fit(~ .,
    data = gene_express_df
  )

hc_fit_ward %>%
  summary()
        Length Class      Mode
spec    4      hier_clust list
fit     7      hclust     list
elapsed 1      -none-     list
preproc 4      -none-     list
hc_fit_ward$fit %>% plot(main=" Ward Linkage
with Correlation - Based Distance ", xlab="", sub ="")

Inference: Yes, the genes separate the samples into the two groups. The results depend on the type of linkage used, but not too much. As we see, we obtain two clusters for average, complete and single, ward linkages except for centroid linkage. Typically, single linkage will tend to yield trailing clusters: very large clusters onto which individual observations attach one-by-one. On the other hand, complete ward and average linkage tend to yield more balanced, attractive clusters. For this reason, complete, ward and average linkage are generally preferred to single and centroid linkage.

2.2 PCA and UMAP (30 pts)

PCA

library(recipes)
library(tidymodels)

# Setup the recipe for PCA
pca_recipe <- recipe(~ ., data = gene_express_df) %>%
  update_role(status, new_role = "ID") %>%  # Mark 'status' as an identifier
  step_normalize(all_predictors(), -all_outcomes()) %>%
  step_pca(all_predictors(), -all_outcomes())

# Prepare the recipe
pca_prep <- prep(pca_recipe)

# Optionally, summarize the prepared recipe
summary(pca_prep)
# A tibble: 6 × 4
  variable type      role      source  
  <chr>    <list>    <chr>     <chr>   
1 status   <chr [3]> ID        original
2 PC1      <chr [2]> predictor derived 
3 PC2      <chr [2]> predictor derived 
4 PC3      <chr [2]> predictor derived 
5 PC4      <chr [2]> predictor derived 
6 PC5      <chr [2]> predictor derived 
library(tidytext)
tidied_pca <- tidy(pca_prep, 2)


tidied_pca %>%
  filter(component %in% paste0("PC", 1:4)) %>%
  group_by(component) %>%
  top_n(8, abs(value)) %>%
  ungroup() %>%
  mutate(terms = reorder_within(terms, abs(value), component)) %>%
  ggplot(aes(abs(value), terms, fill = value > 0)) +
  geom_col() +
  facet_wrap(~component, scales = "free_y") +
  scale_y_reordered() +
  labs(
    x = "Absolute value of contribution",
    y = NULL, fill = "Positive?"
  )

juice(pca_prep) %>%
  ggplot(aes(PC1, PC2, label = status)) +
  geom_point(aes(color = status), alpha = 0.7, size = 2) +
  #geom_text(check_overlap = TRUE, hjust = "inward") +
  labs(x = "Principal Component 1", y = "Principal Component 2", title = "PCA of Gene Expression Data") +
  theme_minimal()

UMAP

library(embed)
library(Matrix)
library(irlba)
umap_rec <- recipe(~., data = gene_express_df) %>%
  update_role(status, new_role = "id") %>%
  step_normalize(all_predictors()) %>%
  step_umap(all_predictors())
umap_prep <- prep(umap_rec)
umap_prep
juice(umap_prep) %>%
  ggplot(aes(UMAP1, UMAP2, label = status)) +
  geom_point(aes(color = status), alpha = 0.7, size = 2) +
#  geom_text(check_overlap = TRUE, hjust = "inward") +
  labs(x = "UMAP1", y = "UMAP2", title = "umap")

2.3 12.6.13 (c) (30 pts)

grp = factor(rep(c(1, 0), each = 20))

regression <- function(y) {
  sum <- summary(lm(y ~ grp))
  pv <- sum$coefficients[2, 4]
  return(pv)
}


out <- tibble(gene = seq(1, nrow(gene_express)),
              p_values = unlist(purrr:: map(1:nrow(gene_express), ~regression(as.matrix(gene_express)[.x, ]))))
out %>% arrange(p_values) %>% head(10)
# A tibble: 10 × 2
    gene p_values
   <int>    <dbl>
 1   502 1.43e-12
 2   589 3.19e-12
 3   600 9.81e-12
 4   590 6.28e-11
 5   565 1.74e-10
 6   551 1.10e- 9
 7   593 1.19e- 9
 8   584 1.65e- 9
 9   538 2.42e- 9
10   569 2.69e- 9
# sig <- out %>% arrange(p_values) %>% filter(p_values < 0.05/nrow(gene_express))
sig <- out %>% arrange(p_values) %>% filter(p_values < 0.05 )

In conclusion, 502, 589, 600,590, 565, 551, 593, 584, 538, 569 are the genes that are significantly different between the two groups (healthy and diseased) as indicated by the adjusted p-values being below 0.05.

# install.packages("pheatmap")
library(pheatmap)
# install.packages("ggplotify")
library(ggplotify) ## to convert pheatmap to ggplot2
# install.packages("heatmaply")
library(heatmaply) ## for constructing interactive heatmap
#create data frame for annotations
dfh <- data.frame(sample=as.character(colnames(gene_express)), status = "disease") %>%
                column_to_rownames("sample")
dfh$status[seq(21, 40)] <-  "healthy"
dfh
      status
ID1  disease
ID2  disease
ID3  disease
ID4  disease
ID5  disease
ID6  disease
ID7  disease
ID8  disease
ID9  disease
ID10 disease
ID11 disease
ID12 disease
ID13 disease
ID14 disease
ID15 disease
ID16 disease
ID17 disease
ID18 disease
ID19 disease
ID20 disease
ID21 healthy
ID22 healthy
ID23 healthy
ID24 healthy
ID25 healthy
ID26 healthy
ID27 healthy
ID28 healthy
ID29 healthy
ID30 healthy
ID31 healthy
ID32 healthy
ID33 healthy
ID34 healthy
ID35 healthy
ID36 healthy
ID37 healthy
ID38 healthy
ID39 healthy
ID40 healthy
pheatmap(gene_express[sig$gene, ], cluster_rows = FALSE, cluster_cols = T, scale="row", annotation_col = dfh,
         annotation_colors=list(status = c(disease = "orange", healthy = "black")),
         color=colorRampPalette(c("navy", "white", "red"))(50))